EQUIVARIANT VECTOR BUNDLES OVER TOPOLOGICAL TORIC MANIFOLDS AND HIRZEBRUCH–RIEMANN–ROCH THEOREM

Abstract

This thesis explores the intersection of algebraic geometry and toric topology, focusing on the classification of equivariant vector bundles over topological toric manifolds and the associated Hirzebruch–Riemann–Roch theorem.

The research is presented in three primary sections. First, we classify complex equivariant vector bundles over topological toric manifolds by synthesizing techniques from differential geometry with conceptual frameworks from algebraic geometry. This work extends Klyachko’s classification of toric vector bundles beyond the realm of smooth toric varieties into the topological setting. Second, we classify real equivariant vector bundles by establishing an equivariant analogue of Atiyah’s equivalence between "Real" vector bundles (complex bundles with a conjugate-linear involution) and real vector bundles. Finally, we provide a geometric interpretation and proof of the combinatorial Hirzebruch–Riemann–Roch theorem, as formulated by Hattori and Masuda, specifically for topological toric manifolds. These results establish new avenues for research in toric topology—most notably regarding the equivariant stability of bundles—and demonstrate the continued utility of algebraic-geometric methods in topological contexts.